Generalized Uncertainty and the WAY Theorem: Intrinsic and Operational Limits on Position Measurement

The continuous and unbounded extension of the Wigner--Araki--Yanase (WAY) theorem by Kuramochi and Tajima establishes a conservation-law obstruction to exact position measurements. We investigate how this result is modified when the measured system obeys a quadratic generalized uncertainty principle (GUP), [X,P] = iℏ(1 +βP2). First, we recover the intrinsic minimal-localization scale ∆Xmin = ℏ√β. Second, assuming an additive conserved momentum, a Yanase-compatible pointer, and the usual hypotheses underlying Ozawa's noise--conservation inequality, we derive a conditional GUP-modified bound on the root-mean-square measurement noise, ϵX ≥ℏ[1 +β⟨P2S⟩]/[2√(∆PS)2 +(∆PA)2]. The result reduces to the canonical bound when β → 0 and makes explicit the dependence on both the system state and apparatus resource. We distinguish this quantitative, model-dependent statement from the intrinsic minimal-length relation and identify the operator-domain and composite-system assumptions required for a full theorem.